 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2014/11/v/solution

All nonzero eigenmodes decay, including the

t e^{K t}

term in the repeated-root case. Total concentration is conserved and initially equals

n

. Combining

A_{\infty} + B_{\infty} + C_{\infty} = n

with the equal-flux ratios gives

A_{\infty} = \frac{n δη}{q}, B_{\infty} = \frac{n ϵδ}{q}, C_{\infty} = \frac{n ϵη}{q}, q = ϵδ + δη + ηϵ .

These are the unique steady state concentrations with this total. The formula remains valid when the transient eigenvalues are complex or repeated.

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